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\begin{center}
  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
\end{center}

\begin{center}
  4:15 p.m., Tuesday, January 17, 2006\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
\end{center}

\begin{center}
  \textsl{Ramani S. Pilla} \\
  Department of Statistics\\
  Department of Biology\\
  Case Western Reserve University \\ 
\end{center}

\begin{center}
  \textbf{Inference in Perturbation Models, Mixtures and Spatial Scan
  Process}
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\noindent In this talk, a general class of models referred to as {\em
perturbation models} are introduced. These models are described by an
underlying ``null'' model that accounts for most of the structure in a
data while a perturbation accounts for possible small localized
departures. For instance, in the context of finite mixture models, the
null model represents a mixture with $m$ components and the
perturbation model represents additional components. In the spatial
scan process context, the null density accounts for the background or
noise whereas the perturbation searches for an unusual region such as
a tumorous tissue in mammography or a target in an image recognition
problem. We derive a new test statistic for detecting the presence of
perturbation and show that the asymptotic distribution of the test
statistic is equivalent to the supremum of a Gaussian process over a
high-dimensional manifold (e.g., curve, surface etc.) with boundaries
and singularities. A technique for approximating the quantiles of the 
test statistic via the Hotelling-Weyl {\em volume-of-tube formula} is
presented.

\noindent Fitting mixture models and performing statistical inference on the
results is an important but a very challenging problem. A long-pending
fundamental question is: {\em how many mixture components}?  The
asymptotic null distribution of the likelihood ratio test statistic is
highly complex and very difficult to simulate from in
practice. Building on the perturbation theory, inferential methods are
developed to address the problem of testing for an arbitrary number of
components from smooth families of distributions, including
multivariate mixtures. The resulting theory has broad applications
including astronomy, astrophysics, particle physics, bioinformatics
and genetics. We illustrate the theory in the context of a model
problem from high-energy particle physics wherein the goal is to
distinguish a signal from random fluctuation in data with a high
probability. More information on the particle physics and other
application problems is at {\sf http://stat.case.edu/$\sim$pillar/PRL/PRL.htm}.

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